Am I an asshole for excluding a girl from my birthday celebration?

My birthday was recently and I decided to celebrate with my college friends by going to a party today (Friday), and I only invited my close friends (6 girls), but in college our group actually has 7 people.

Let's call her L, L is a girl who has been annoying me since last semester, she actually hangs out with our group of friends because she is friends with one person in the group, just one. L invites herself to do all of our college work with us, and she annoys everyone EVERY TIME, she is always a disservice and she even missed one of the presentations once. She also always says something nasty, she badmouths everyone and even tried to badmouth a person in our group once, she has also said homophobic things about a girl in our class. I don't know, I just don't like her. I've never treated her badly, even though I've reprimanded her when she's been homophobic, I've never been rude to her or had the courage to push her away. Honestly, it seems like the problem is exactly that: no one likes her and NO ONE has the courage to say anything.

The situation is this: I didn't invite her to go to the party with us today and she heard two of my friends talking about it. When L asked what they were talking about, my friend was embarrassed to tell her about a party L clearly was the only one who wasn’t invited and ended up inviting her too.

L sent a message to this friend after class, asking for details about the place and time we were going, and my friend simply told L to talk to me about the details of the party, because after all I was the one who invited her. L then said that she thought we were just going to a party, for no reason, and that she wouldn't go because I hadn't invited her.

I thought that was settled, but then hours later L texts me asking if she could come with us to the party. What do I do? Do I leave her on read or politely reply saying no? Or do I invite her to go with us? Honestly, I don't like her, we're not close, and she didn't even wish me a happy birthday. Am I being a jerk?